A Generalization of QR Factorization To Non-Euclidean Norms

01/24/2021
โˆ™
by   Reid Atcheson, et al.
โˆ™
0
โˆ™

I propose a way to use non-Euclidean norms to formulate a QR-like factorization which can unlock interesting and potentially useful properties of non-Euclidean norms - for example the ability of l^1 norm to suppresss outliers or promote sparsity. A classic QR factorization of a matrix ๐€ computes an upper triangular matrix ๐‘ and orthogonal matrix ๐ such that ๐€ = ๐๐‘. To generalize this factorization to a non-Euclidean norm ยท I relax the orthogonality requirement for ๐ and instead require it have condition number ฮบ ( ๐ ) = ๐ ^-1๐ that is bounded independently of ๐€. I present the algorithm for computing ๐ and ๐‘ and prove that this algorithm results in ๐ with the desired properties. I also prove that this algorithm generalizes classic QR factorization in the sense that when the norm is chosen to be Euclidean: ยท=ยท_2 then ๐ is orthogonal. Finally I present numerical results confirming mathematical results with l^1 and l^โˆž norms. I supply Python code for experimentation.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
โˆ™ 05/15/2023

Sparsifying Sums of Norms

For any norms N_1,โ€ฆ,N_m on โ„^n and N(x) := N_1(x)+โ‹ฏ+N_m(x), we show ther...
research
โˆ™ 05/13/2020

Implicit Regularization in Deep Learning May Not Be Explainable by Norms

Mathematically characterizing the implicit regularization induced by gra...
research
โˆ™ 03/18/2021

An algorithm for J-spectral factorization of certain matrix functions

The problems of matrix spectral factorization and J-spectral factorizati...
research
โˆ™ 08/31/2021

The Approximation Ratio of the k-Opt Heuristic for the Euclidean Traveling Salesman Problem

The k-Opt heuristic is a simple improvement heuristic for the Traveling ...
research
โˆ™ 02/06/2020

Computing the norm of nonnegative matrices and the log-Sobolev constant of Markov chains

We analyze the global convergence of the power iterates for the computat...
research
โˆ™ 01/17/2019

Efficient Matrix Profile Computation Using Different Distance Functions

Matrix profile has been recently proposed as a promising technique to th...
research
โˆ™ 10/12/2019

New robust ScaLAPACK routine for computing the QR factorization with column pivoting

In this note we describe two modifications of the ScaLAPACK subroutines ...

Please sign up or login with your details

Forgot password? Click here to reset